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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<a NAME="E1:3"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">c<sub>1</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V3">discerning</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V4">symmetric</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V5">triangle</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:3_1"/><b>then </b><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<a NAME="E3:3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="metric_6.html#E1:3_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V2">Reflexive</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D3">METRIC_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E5:3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V3">discerning</a>
 <b>by </b><i><a class="txt" href="metric_6.html#E2:3_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="metric_1.html#D4">METRIC_1:def 4</a></i>;<br/>

<a NAME="E6:3_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E1:3_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V4">symmetric</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D5">METRIC_1:def 5</a></i>;<br/>

<a NAME="E8:3_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="metric_6.html#E1:3_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V5">triangle</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D6">METRIC_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:3"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V3">discerning</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V4">symmetric</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V5">triangle</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:3_2"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V3">discerning</a> )
 <b>and </b><br/><a NAME="E2:3_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V4">symmetric</a>
 <b>and </b><br/><a NAME="E3:3_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V5">triangle</a>
 ;<br/>

<a NAME="E4:3_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="metric_6.html#E1:3_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="metric_6.html#E2:3_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="metric_6.html#E3:3_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="metric_1.html#D3">METRIC_1:def 3</a>, <a class="ref" href="metric_1.html#D4">METRIC_1:def 4</a>, <a class="ref" href="metric_1.html#D5">METRIC_1:def 5</a>, <a class="ref" href="metric_1.html#D6">METRIC_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="pcomps_1.html#R1">is_metric_of</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V3">discerning</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V4">symmetric</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="metric_1.html#V5">triangle</a> ) )
 <b>by </b><i><a class="txt" href="metric_6.html#E1:3"><i><font color="Green">E4</font></i></a></i>;<br/>


</div>
